Three solutions for a nonlocal problem with critical growth

Abstract: The main goal of this work is to prove the existence of three different solutions (one positive, one negative and one with nonconstant sign) for the equation $(-\Delta_p)^s u= |u|^{p^{*}_s -2} u +\lambda f(x,u)$ in a bounded domain with Dirichlet condition, where $(-\Delta_p)^s$ is the well known $p$-fractional Laplacian and $p^*_s=\frac{np}{n-sp}$ is the critical Sobolev exponent for the non local case. The proof is based in the extension of the Concentration Compactness Principle for the $p$-fractional Laplacian and Ekeland's variational Principle.